

Here the superscripts related to the different frames have been
omitted for notational brevity. Further, \(t\)  and \(T\)  denote the time
and discretization (sample)
period, respectively. Moreover, \(\mathcal {O}(T^n)\)  denotes \(n\) :th and
higher-order terms of \(T\) . However, integration by Taylor
expansion is not possible for the rotation matrix \(R\) , since it
belongs to \(SO(3) = \lbrace  R \in \operatorname{\mathbb {R}}^{3 \times 3}: R^\top R = I_3, \det R= 1 \rbrace \) . That is, the orthogonality and determinant constraints
impose \(SO(3)\)  to be a 3-dimensional manifold embedded in \(\operatorname{\mathbb {R}}^{3\times 3}\) . However, noting that \(SO(3)\) 
is a matrix Lie group, Lie theory can be employed to integrate \(R\)  and
discretize
(REF ). Instead of
integrating in \(\operatorname{\mathbb {R}}^{3 \times 3}\) , the integration is performed in the
tangent space to \(R\) , also called the Lie algebra, which is isomorphic
to \(\operatorname{\mathbb {R}}^3\) . For sufficiently small \(T\)  the solution
to (REF ) is [1]}
\(R(t+T) = R(t) \exp _{SO(3)}^{\wedge }(\theta (T) )\) 
